We study the second temporal Rényi entropy S₂ = −log Tr ρ_A² of a heavy primary state in a 2d CFT, for an interval of temporal extent Δt and spatial extent Δφ. For BTZ microstates we define a gravitational model F_grav, built from the Gaussian-ensemble average over heavy OPE data: a sum of exact finite-c Virasoro vacuum blocks, continued to the Lorentzian sheet, in which the integer-spin projection (a sum over Dehn-twist images) is truncated to five images. Within F_grav, Tr ρ_A² has zeros at real (Δt, Δφ) near the transition between the same-sheet and winding saddles. All 18 zeros found for 46 ≤ c ≤ 241 pass a numerical validation by a degree argument. For the zeros at c = 46 and c = 61 we give computer-assisted proofs. Interval arithmetic with analytic bounds on every omitted series term shows that F_grav has Brouwer degree −1 on a disk of radius 0.01 around each. Analytic bounds on all omitted images show that on each disk the complete image sum differs from F_grav by less than 1/160 of a rigorous lower bound on |F_grav| at the boundary, so it also has a zero there. Assuming mainly that only two saddles compete near the transition, that their large-c expansion is uniform there and that their relative phase is monotonic, the zeros follow a Lee-Yang law. We test it numerically: it predicts their number in a fixed window and their positions out of sample, to within 0.0033 in Δφ. The spacing of consecutive zeros, which requires only |Tr ρ_A²|, gives the modulus of the mean slope of the relative action phase between them to within 2%. Persistence of the zeros in individual microstates requires a bound on the deviation of the microstate correlator from F_grav, or at c = 46 and c = 61 from the complete image sum, which we do not establish. Files. Robles_Blanquer_Fisher_zeros_BTZ.pdf is the manuscript. ancillary_files.zip contains its ancillary files (the folder anc/). certificate_c46/ and certificate_c61/ contain the programs, the coefficient data and the stored interval certificates of the computer-assisted proofs of Theorem 1 (c = 46) and Theorem 2 (c = 61), with checkers, pinned requirements and SHA-256 manifests. independent_checks/ contains independent checks, which are not inputs to the proofs. The other files are the numerical pipeline of the paper, the series coefficients, the validation records and the figure scripts. README.md lists the script of each step, table, figure and check. To check the stored certificates, use Python 3.12 with the pinned requirements of each folder (python-flint 0.9.0, mpmath 1.4.1, numpy 2.3.5, scipy 1.17.0). From the folder where the archive is unpacked, run:cd anc/certificate_c46/F6_boundary && python work/check_certificate.pycd anc/certificate_c46/image_sum && python work/check_image_tail.pycd anc/certificate_c61 && python work/check_image_tail.py 61 This preprint has not been peer reviewed. The computations, the code, the computer-assisted proofs and the manuscript were developed with substantial assistance from AI systems, under the author's direction. The author takes full responsibility for the content.
We study the second temporal Rényi entropy S₂ = −log Tr ρ_A² of a heavy primary state in a 2d CFT, for an interval of temporal extent Δt and spatial extent Δφ. For BTZ microstates we define a gravitational model F_grav, built from the Gaussian-ensemble average over heavy OPE data: a sum of exact finite-c Virasoro vacuum blocks, continued to the Lorentzian sheet, in which the integer-spin projection (a sum over Dehn-twist images) is truncated to five images. Within F_grav, Tr ρ_A² has zeros at real (Δt, Δφ) near the transition between the same-sheet and winding saddles. All 18 zeros found for 46 ≤ c ≤ 241 pass a numerical validation by a degree argument. For the zeros at c = 46 and c = 61 we give computer-assisted proofs. Interval arithmetic with analytic bounds on every omitted series term shows that F_grav has Brouwer degree −1 on a disk of radius 0.01 around each. Analytic bounds on all omitted images show that on each disk the complete image sum differs from F_grav by less than 1/160 of a rigorous lower bound on |F_grav| at the boundary, so it also has a zero there. Assuming mainly that only two saddles compete near the transition, that their large-c expansion is uniform there and that their relative phase is monotonic, the zeros follow a Lee-Yang law. We test it numerically: it predicts their number in a fixed window and their positions out of sample, to within 0.0033 in Δφ. The spacing of consecutive zeros, which requires only |Tr ρ_A²|, gives the modulus of the mean slope of the relative action phase between them to within 2%. Persistence of the zeros in individual microstates requires a bound on the deviation of the microstate correlator from F_grav, or at c = 46 and c = 61 from the complete image sum, which we do not establish. Files. Robles_Blanquer_Fisher_zeros_BTZ.pdf is the manuscript. ancillary_files.zip contains its ancillary files (the folder anc/). certificate_c46/ and certificate_c61/ contain the programs, the coefficient data and the stored interval certificates of the computer-assisted proofs of Theorem 1 (c = 46) and Theorem 2 (c = 61), with checkers, pinned requirements and SHA-256 manifests. independent_checks/ contains independent checks, which are not inputs to the proofs. The other files are the numerical pipeline of the paper, the series coefficients, the validation records and the figure scripts. README.md lists the script of each step, table, figure and check. To check the stored certificates, use Python 3.12 with the pinned requirements of each folder (python-flint 0.9.0, mpmath 1.4.1, numpy 2.3.5, scipy 1.17.0). From the folder where the archive is unpacked, run:cd anc/certificate_c46/F6_boundary && python work/check_certificate.pycd anc/certificate_c46/image_sum && python work/check_image_tail.pycd anc/certificate_c61 && python work/check_image_tail.py 61 This preprint has not been peer reviewed. The computations, the code, the computer-assisted proofs and the manuscript were developed with substantial assistance from AI systems, under the author's direction. The author takes full responsibility for the content.
We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by Δ . For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x 2 Δ +2 . For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q =exp(−2π β /L), with β being the inverse temperature and L being the spatial period, to order q Δ +2 . To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.
We prove an exact reflection of the energy–splitting spectral curves for a class of Rabi–Stark Hamiltonians. Under common-domain, semiboundedness, and compact-resolvent assumptions, it preserves eigenvalue order and multiplicity without requiring the bosonic terms to commute. Its fixed line recovers known one-photon degeneracies. For two-photon coupling, reflected simple states are orthogonal and their marginal trace distances are explicit. A quantitative contact limit then yields universal probabilities for fixed detected counts near collapse. At known fixed efficiency, one- and two-count proportions distinguish the ground-state branches despite unknown source magnitude and missed acquisitions: consistency is possible precisely when M√Ω → ∞. With independent Poisson background of unknown positive mean and a positive missed-acquisition fraction, counts zero through four and overflow suffice, with threshold MΩ → ∞. Two exact minors establish the readout's rank at every interior efficiency and identify the leading blind point when count four is omitted. Here M is the total preparation budget and Ω ↓ 0 is the collapse limit. Generative-AI disclosure and author responsibility. ChatGPT (GPT-6 Astra Pro) was used for mathematical exploration, proof checking, drafting, and preparation of verification programs. The author is responsible for the final manuscript. 2020 Mathematics Subject Classification: Primary 81Q10; Secondary 81V80, 62F03. Files: the compiled manuscript Rabi_Stark_Spectral_Reflection_v1.pdf (14 pages) and the source archive Rabi_Stark_Spectral_Reflection_v1_source.zip, which contains the self-contained LaTeX source main.tex (bibliography included; compiles with pdfLaTeX) and the ancillary directory anc/: Python verification and readout programs, the stored original-Hamiltonian probability records behind the tables, the outputs of the review checks, a README, and SHA256SUMS.txt for package integrity. Running python anc/verify.py (Python 3.10 or later; dependencies in anc/requirements.txt) reruns the exact symbolic spin and determinant checks, independent coefficient sums, tensor-product spectral reflection tests, and the readout on the stored records; anc/verify_general_reflection.py checks the general reflection on 44 spectral pairs, and anc/reproduce.py recomputes the original-Hamiltonian count laws. Numerical checks are floating-point diagnostics, not interval certificates.
We extend and refine recent results on Rényi entropy in two-dimensional con-formal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and sub-leading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Rényi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.
We prove the ground-state energy of the periodic, half-filled 4×4 repulsive Hubbard model with t = 1 to at least 23 decimal places at U = 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 and 16: every enclosure has width below 10⁻²⁴. The double occupancy, the spin correlations and the staggered structure factor have certified interval widths below 10⁻¹⁰, and the kinetic energy below 10⁻⁹. Hypercube, spin and pseudospin symmetries reduce the problem to a 19,186-dimensional block that contains the ground state by Lieb's theorem and a character argument. A binary64 Cholesky factorization, verified using a rigorous rounding-error bound evaluated in exact rational arithmetic, certifies spectral separation within this block; exact Rayleigh quotients and residuals then give Temple enclosures of the energy and bounds on the angle to the ground state, which certify the observables. Separation certificates at both ends and a perturbative trial family enclose E₀(U) for every U in [3.9, 4.1], with uniform width at most 1.6 × 10⁻⁹. Two independent implementations replay the certificates, the interlacing and Temple steps are proved in Lean 4, and all computations run on one laptop. At U = 4 and 8, exact-arithmetic brackets that use neither Lieb's theorem nor floating point are also given, and the enclosures bound their deficits. At U = 4 the certified energy rounds to −13.62185 and excludes the published value −13.62192 [Physical Review B 94, 085103]. Contents of the bundle paper/ — the manuscript (Markdown and PDF), its figures, and supplementary/ with five videos generated from the certificate files. rayleigh/symmetry/ — the enclosure pipeline (block dimensions, exact basis, certificates, observables, the interval certificate) and recheck/, the independent Rust reference checker; certificates/ holds every certificate, trial vector, run log, checker log and the clean-replay report. sos/, rayleigh/demo/, cluster/ — the exact-arithmetic bracket certificates and the expected outputs of their checkers (golden_ files); the bracket checkers belong to the authors' certification software and are not included. formal/ — the Lean 4 statements (rank-one interlacing, Temple's inequality, the sum-of-squares remainder bound). tests/ — the generators of every table, figure and video of the manuscript. replay.sh, REPLAY.md — the automated replay and the step-by-step commands. AUDIT_AXIOMS.txt — #print axioms of the formal statements. SHA256SUMS — checksums of every file. Verification PYTHON=<python with numpy and scipy> ./replay.sh rebuilds the symmetry block from the Hamiltonian, reproduces every certificate from the stored integer vectors, runs the Rust reference checker, recomputes the observables and checks the manuscript's tables, and writes REPLAY_REPORT.md. It needs about 12 GB of memory and 16 GB of free disk, and takes about half an hour on the hardware above; its work directory is removed at the end. REPLAY.md lists the commands step by step; rayleigh/symmetry/certificates/REPLAY_REPORT_2026-10-14.md is a clean replay of this material. Not included: the two 662,547,600-byte full-sector vectors of the upper bracket certificates (their SHA-256 digests are in rayleigh/demo/). Integrity: shasum -a 256 -c SHA256SUMS inside the bundle; the SHA-256 of the tarball is in the .sha256 file beside it. Environment Python: 3.13.7 with numpy 2.5.4, scipy 1.18.1, matplotlib 3.11.2 (producer-requirements.txt) Rust: 1.90.0 (reference checker; cargo build --release in rayleigh/symmetry/recheck) Lean: leanprover/lean4:v4.33.1 with Mathlib v4.33.1 (formal statements) Hardware: Apple M3 Pro, 18 GiB (all production and replay) License: CC BY-NC 4.0. Commercial use requires written permission from Origin Neural AI (labs@smartledger.solutions).
Preprint and reproducibility archive; not peer reviewed. What this record contains. The full text of the manuscript "Identical Few-Particle Gauge Hamiltonians, Programmable Collective Charge Response" (M. J. Baek), its Supplementary Information with complete proofs, and the archive that certifies every closed-form value the paper states. The result. Few-particle dynamics are routinely used to infer collective response in paired quantum matter. This work constructs finite-range interacting response twins that defeat that inference exactly. For any prescribed finite p, two positive finite-range parents have complete static gauge Hamiltonians that coincide at every Peierls link phase on every sector N ≤ p and first separate at N = p+1; their zero-twist finite-density ground rays are the same Dicke condensate, yet their collective charge curvatures differ extensively. The obstruction has no universal depth, and the paper locates it exactly: the blind depth equals the spectator body order, the algebraic bound is independent of geometric range, and bounded range still bounds depth only through the number of sites a ball can hold — so no arbitrary-depth family of uniformly fixed range is claimed. Conversely, a structurally closed exact-pair tower closes at depth two. What is hidden is designable: with d(d+1)/2 displacement families the whole symmetric response-tensor difference is locally realizable at interior filling, and a triangular-lattice twin pair differs only in shear while both diagonal responses stay equal. A 4×4 hard-core-qubit discriminator gives an exact factor-of-two contrast with an internal null control and a state-preparation certificate in 17 equal-time settings. What is certified, and what is not. Results proved in closed form are stated as theorems and proved in full in the Supplementary Information; results established by exact rational or symbolic computation are labelled computer-assisted and carry the script that produces them; the 16-qubit discriminator is a blueprint with an end-to-end noise simulation, not a hardware demonstration, which is not claimed anywhere. Scope limits, open proof lines and one refuted conjecture are recorded explicitly rather than omitted. The archive. response_twins_unified_v1.0.zip contains the manuscript and Supplementary sources, the proof notes, the complete fail-closed verification harness (31 executed stages in release mode — 27 claim-bearing certificate stages, 1 non-claim diagnostic stage and three fail-closed gates — and 24 in quick mode) with one frozen log per stage, independent reimplementations of the load-bearing calculations, negative controls that must fail, the exact-arithmetic certificates behind every closed-form value in the paper, the figure-generation scripts, the scientific claim and provenance record (79 rows with status and scope notes, 50 of them naming the in-archive script or certificate that checks them, the rest saying in words why not), and a per-file SHA-256 manifest of all 209 payload files. Verifying it. Extract the archive and run python3 REPRODUCIBILITY/run_all.py --mode quick for the fast subset or --mode release for all 31 stages; bash BUILD.sh rebuilds both PDFs from source. The three gates check journal-format limits, the release payload against its manifest, and every number the documentation states about the artefacts. Running the harness does not modify the archive: all runtime output is written outside the immutable tree, so verification cannot invalidate the manifest. README.md separates reproducing the content (TeX Live 2023 or newer) from reproducing the archived PDF bytes, which needs the exact producing build recorded inside the PDFs. Licences. Manuscript, Supplementary Information and documentation CC BY 4.0; source code MIT; certificate data CC0 1.0. LICENSE.md is the authoritative per-file mapping and is included both in the archive and as a separate file on this record.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.